# the Catalan's semi-regular polyhedra

These semi-regular polyhedra of the second kind are the duals of the Archimedes' polyhedra. Their faces are superimposable (but not regular) and tangent to a sphere (inscribed sphere); their vertices are regular (of two or three types).

Each drawing provides a link to a popup applet (more about the characteristics of the semi-regular polyhedra).

The dual of the cuboctahedron (resp. icosidodecahedron) has 12 (resp. 30) rhombic faces.
Its vertices of order 3 are the vertices of a cube (resp. a regular dodecahedron); the remaining vertices are those of a regular octahedron (resp. icosahedron).
Thus these polyhedra can be made by assembling regular pyramids with a square (resp. pentagonal) base on each face of a cube (resp. a regular dodecahedron), or by assembling tetrahedra on the faces of a regular octahedron (resp. icosahedron).
 dodécaèdre rhombique animations rhombic triacontahedron (Romé de l'Isle) Remark (see the "animations"): the six pyramids to assemble on the cube are the sixths of the cube, thus the rhombic dodecahedron fills the space (we get this paving by dissecting one of two cubes in a cubic paving).

The duals of the snub cube and of the snub dodecahedron have respectively 24 and 60 pentagonal faces.
 pentagonal icotetrahedron pentagonal hexecontahedron

exercise (difficult): In a unit cube we consider the three edges stemming from a same vertex; the ends of two of them and the midpoint of the third define un plane along which we cut a tetrahedron. We carry out the 24 cuts along such planes. Describe the resulting polyhedron; what is its volume?
answers: the "24 kites", its half-net (to build your own) and its volume.

# the diamonds and the antidiamonds (dipyramids and trapezohedra)

These duals of the prisms and of the antiprisms complete the list of the second kind semiregular polyhedra.
(The diamond of order 4 is the regular octahedron; the antidiamond of order 3 is the cube.)

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